Wasserstein Distance in Cosmological Structure Formation: An Optimal Transport Perspective

Abstract

The formation of cosmological large-scale structure is usually described in terms of the evolution of density fluctuations and their statistical measures, such as the power spectrum and correlation function. However, these statistics characterize the amplitude structure of density fluctuations and do not directly describe the spatial redistribution of matter that occurs during structure formation. In this work we formulate cosmological structure formation as a transport problem of mass distributions using the Wasserstein distance from optimal transport theory. The generative process from the initial linear density field to the observed galaxy catalog is treated as a hierarchical mapping from a continuous density field to a galaxy point process, and an approximate expression for the Wasserstein distance between them is derived under the small-fluctuation approximation. We show that this distance naturally decomposes into contributions associated with three physical processes: mass transport by gravitational evolution, galaxy formation bias, and shot noise arising from the discrete sampling of galaxies. The gravitational transport term is expressed as an integral of the matter power spectrum, while the galaxy formation contribution appears as a weighted integral of the galaxy correlation function. The sampling term corresponds to Poisson shot noise originating from the discreteness of the galaxy catalog. These results provide a unified framework for describing cosmological large-scale structure formation from the perspective of transport geometry and suggest that the Wasserstein distance may serve as a new statistical quantity linking continuous density fields with observed galaxy catalogs.